Let $f_n$: $[0,1]\to\mathbb{R}$ be differentiable satisfying both:
$$\begin{align}
(1)& \int_0^1f_n(x)dx=0,\quad \forall n\in\mathbb{N},\\
(2)& |f'_n(x)|\le \frac{1}{\sqrt{x}},\quad\forall x\in(0,1].
\end{align}$$
Prove that $f_n$ has a uniformly convergent subsequence on $[0,1]$.
I want to apply Arzela-Ascoli Theorem, so I try to show that $f_n$ is uniformly bounded and equicontinuous. But I only get that $f_n$ is pointwise bounded. And I have no idea how to use the first condition.
Best Answer
Note that since $f'_n$ is not assumed continuous, we can't apply directly fundamental theorem of analysis. But we can use the fact that $f_n$ is absolutely continuous, with a good condition on the derivative, to get what we want.